Reduced Krylov Basis Methods for Parametric Partial Differential Equations
This talk presents a user-friendly reduced basis method for solving a family of parametric PDEs by preconditioned Krylov subspace methods including the CG method, GMRes, and BiCGStab.
Overview
The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then the large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. As shown in the theory and experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale systems in the reduced basis subspace. This reduces the computational cost dramatically because (1) to construct the reduced basis vectors, we only solve one large-scale problem on the high-fidelity level; and (2) the problems for any value in the parameter set have much smaller dimensions they are restricted to the subspace defined during the Krylov iterations. This is a joint work with Yuwen Li (Zhejiang University) and Cheng Zuo (Penn State).
Presenters
Ludmil Zikatanov, Professor, Mathematics, Pennsylvania State University
Brief Biography
Ludmil Zikatanov is a Full Professor in the Department of Mathematics at The Pennsylvania State University. He earned his Ph.D. in Mathematics from Sofia University “St. Kliment Ohridski” in Sofia, Bulgaria.
His research spans the broad field of computational mathematics, with particular emphasis on numerical methods for partial differential equations, numerical linear algebra, and multilevel methods for linear systems and optimization. He is also interested in nonlinear approximation techniques for data compression and machine learning, as well as the application of these mathematical methods in hydrogeology, resource economics, physics, and other scientific disciplines.
